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Archive for Mathematical Logic

, Volume 56, Issue 3–4, pp 187–213 | Cite as

Chains of saturated models in AECs

  • Will Boney
  • Sebastien VaseyEmail author
Article

Abstract

We study when a union of saturated models is saturated in the framework of tame abstract elementary classes (AECs) with amalgamation. We prove:

Theorem 0.1. If K is a tame AEC with amalgamation satisfying a natural definition of superstability (which follows from categoricity in a high-enough cardinal), then for all high-enough \(\lambda {:}\)
  1. (1)

    The union of an increasing chain of \(\lambda \)-saturated models is \(\lambda \)-saturated.

     
  2. (2)

    There exists a type-full good \(\lambda \) -frame with underlying class the saturated models of size \(\lambda \).

     
  3. (3)

    There exists a unique limit model of size \(\lambda \).

     
Our proofs use independence calculus and a generalization of averages to this non first-order context.

Keywords

Abstract elementary classes Forking Independence calculus Classification theory Stability Superstability Tameness Saturated models Limit models Averages Stability theory inside a model 

Mathematics Subject Classification

Primary 03C48 Secondary 03C47 03C52 03C55 03E55 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA
  2. 2.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA

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